Finite Group Databasebeta

← #6564404

Subgroups — #6564404

Order 512.

1957 conjugacy classes of subgroups.

The subgroup lattice has 694 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3333C212yesmaximal
2#36180C211yesmaximal, characteristic, complemented
3#34330C211yesmaximal, characteristic
4#29869C211yesmaximal, characteristic, complemented
5#29717C212yesmaximal, complemented
6#28716C211yesmaximal, characteristic, complemented
7#28577C212yesmaximal, complemented
8#28160C211yesmaximal, characteristic, complemented
9#27644C212yesmaximal, complemented
10#25322C212yesmaximal, complemented
11#25223C214yesmaximal, complemented
12#24682C212yesmaximal, complemented
13#24548C214yesmaximal, complemented
14#23679C211yesmaximal, characteristic, complemented
15#23630C212yesmaximal, complemented
16#22700C211yesmaximal, characteristic, complemented
17#22502C212yesmaximal, complemented
18C2×#554C2212yes—
19C2×#550C2211yescharacteristic
20#3333—216no—
21C2×#486C2212yes—
22C2×#480C2214yes—
23C2×#477C2212yes—
24C2×#468C2214yes—
25C2×#460C2212yes—
26C2×#452C2214yes—
27C2×C4×Q16C2212yes—
28C2×#436C2212yes—
29C2×#417C2214yes—
30#2868C2211yescharacteristic
31#2857C2212yes—
32#2567C2212yes—
33#2552—21no—
34#2552—21no—
35#2502C2214yes—
36#2440—22no—
37#2440—22no—
38#2366—21no—
39#2366C2211yescharacteristic
40#2364—22no—
41#2363C2212yes—
42#2357—21no—
43#2355C2211yescharacteristic
44#2354C2211yescharacteristic
45D4×C4⋊C4—21no—
46D4×C4⋊C4—21no—
47#2339C2212yes—
48#2333—22no—
49#2328—21no—
50C4×C4⋊Q8C2211yescharacteristic
51C4×#127C2211yescharacteristic
52C4×#121C2212yes—
53#2297C2212yes—
54#2295C2211yescharacteristic
55C2×#384C2212yes—
56#2275—21no—
57C42×D4C2211yescharacteristic
58#2083C2211yescharacteristic
59#2083C2212yes—
60#2083C2212yes—
61#2072C2212yes—
62#2072C2211yescharacteristic
63#2072C2214yes—
64#2072C2212yes—
65#2071C2212yes—
66#2071C2212yes—
67#2071C2212yes—
68#2071C2214yes—
69#2068C2212yes—
70#2056C2214yes—
71#2056C2212yes—
72#2055C2214yes—
73#2043C2211yescharacteristic
74#2016C2211yescharacteristic
75#2013C2212yes—
76#1990C2212yes—
77#1987C2212yes—
78#1947C2214yes—
79#1939C2212yes—
80#1928C2214yes—
81#1894C2214yes—
82#1877C2214yes—
83#1877C2214yes—
84#1873C2214yes—
85#1851C2214yes—
86#1831C2211yescharacteristic
87#1823C2214yes—
88#1803C2212yes—
89#1802C2214yes—
90#1802C2214yes—
91#1797C2212yes—
92C4×C4⋊C8C2211yescharacteristic
93C4×#103C2212yes—
94#1736C2214yes—
95#1728C2212yes—
96C22×#41C2312yes—
97C22×Q16C2312yes—
98C22×SD16C2314yes—
99C22×M16C2314yes—
100#554—28no—
101#554—28no—
102#550—24no—
103#550D418yes—
104C2×C4⋊Q8—21no—
105C2×C4⋊Q8—21no—
106C2×C4⋊Q8C2311yescharacteristic
107C2×C4⋊Q8C2312yes—
108C2×#127—21no—
109C2×#127C2311yescharacteristic
110C2×#125—22no—
111C2×C22⋊Q8—24no—
112C2×C22⋊Q8C2314yes—
113C2×#121—22no—
114C2×#121C2312yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2311yescharacteristic
117C2×C4×D4—22no—
118C2×C4×D4—22no—
119C2×C4×D4—21no—
120C2×C4×D4—21no—
121C2×C4×D4—21no—
122C2×C4×D4—21no—
123C2×C4×D4D411yescharacteristic
124C2×C4×D4D411yescharacteristic
125C2×C4×D4C2312yes—
126C2×C4×D4C2312yes—
127C2×C4×D4C2311yescharacteristic
128C2×#117C2312yes—
129C22×C4⋊C4—22no—
130C22×C4⋊C4—22no—
131C22×C4⋊C4C2314yes—
132C22×C42C2312yes—
133#486—28no—
134#480—216no—
135#477—28no—
136#468—216no—
137#460—28no—
138#452—216no—
139C4×Q16—28no—
140#436—28no—
141C2×#107C2314yes—
142C2×#107C2312yes—
143C2×C4⋊C8C2312yes—
144C2×C4⋊C8C2314yes—
145C2×C4⋊C8C2312yes—
146#417—216no—
147C2×#103C2314yes—
148C2×#103C2312yes—
149C2×#103C2312yes—
150C2×#103C2312yes—
151C2×#103C2314yes—
152C2×#103C2312yes—
153C2×#102C2314yes—
154C2×#102C2314yes—
155C2×#102C2314yes—
156C2×C22⋊C8C2314yes—
157C2×C4×C8C2312yes—
158#398—24no—
159#398—24no—
160#395—24no—
161#391C2312yes—
162#391C2314yes—
163#391C2311yescharacteristic
164#391C2312yes—
165#391C2311yescharacteristic
166#390—21no—
167#390—21no—
168#390—22no—
169#390—21no—
170#390—21no—
171#390C2312yes—
172#390C2312yes—
173#389—22no—
174#389—22no—
175#389—21no—
176#389—21no—
177#389—22no—
178#389—22no—
179#389C2311yescharacteristic
180#389C2311yescharacteristic
181#388—22no—
182#388—22no—
183#388—22no—
184#387—21no—
185#387—21no—
186#387—22no—
187#386—22no—
188#386—22no—
189#386—22no—
190#386—22no—
191#386—22no—
192#386—22no—
193#386—22no—
194#385—24no—
195#385—24no—
196#384—21no—
197#384—21no—
198#384—22no—
199#384—22no—
200#384C2311yescharacteristic
201#384C2311yescharacteristic
202#384C2314yes—
203#384C2312yes—
204#384C2312yes—
205#382—21no—
206#381—22no—
207#380—22no—
208#380—22no—
209#380C2312yes—
210C4×C4⋊C4—21no—
211C4×C4⋊C4—21no—
212C4×C4⋊C4—21no—
213C4×C4⋊C4—21no—
214C4×C4⋊C4—21no—
215C4×C4⋊C4D411yescharacteristic
216C4×C4⋊C4D411yescharacteristic
217C4×C4⋊C4C2311yescharacteristic
218C4×C4⋊C4C2312yes—
219C4×C4⋊C4C2311yescharacteristic
220C4×C22⋊C4—22no—
221C4×C22⋊C4—22no—
222C4×C22⋊C4—22no—
223C4×C22⋊C4—22no—
224C4×C22⋊C4D412yes—
225C4×C22⋊C4D412yes—
226C4×C22⋊C4C2312yes—
227#376—21no—
228C2×#95—22no—
229C43C2311yescharacteristic
230#340C2314yes—
231#340C2312yes—
232#340C2314yes—
233#340C2314yes—
234#340C2314yes—
235#340C2314yes—
236#340C2312yes—
237#336C2311yescharacteristic
238#336C2312yes—
239#336C2311yescharacteristic
240#336C2314yes—
241#336C2312yes—
242C2×#41—24no—
243C2×#41—44no—
244C2×#41—44no—
245C2×#41C2×D418yes—
246C22×Q8—22no—
247C22×Q8C2412yes—
248C22×Q8C2411yescharacteristic
249C22×D4—22no—
250C22×D4—22no—
251C22×D4—21no—
252C22×D4—21no—
253C22×D4C2×D411yescharacteristic
254C22×D4C2412yes—
255C23×C4—22no—
256C23×C4—24no—
257C23×C4—22no—
258C23×C4—22no—
259C23×C4C2×D412yes—
260C23×C4C2×D412yes—
261C23×C4C2412yes—
262C2×Q16—24no—
263C2×Q16—42no—
264C2×Q16—42no—
265C2×SD16—28no—
266C2×SD16—44no—
267C2×SD16—44no—
268C2×M16—28no—
269C2×M16—44no—
270C2×M16—44no—
271C22×C8C2412yes—
272C22×C8C2414yes—
273C22×C8C2412yes—
274C4⋊Q8—24no—
275C4⋊Q8—24no—
276C4⋊Q8—42no—
277C4⋊Q8—82no—
278C4⋊Q8—82no—
279#127—28no—
280#127—22no—
281#127C2×D414yes—
282#125—48no—
283C22⋊Q8—28no—
284C22⋊Q8—28no—
285C22⋊Q8—416no—
286#121—24no—
287#121—216no—
288#121C2×D418yes—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8C2×D414yes—
293C4×D4—28no—
294C4×D4—24no—
295C4×D4—24no—
296C4×D4—44no—
297C4×D4—44no—
298C4×D4—48no—
299C4×D4—44no—
300C4×D4—44no—
301C4×D4—82no—
302C4×D4—84no—
303C4×D4C2×D414yes—
304C4×D4#4114yes—
305#117—44no—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—28no—
308C2×C4⋊C4—28no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—21no—
311C2×C4⋊C4—22no—
312C2×C4⋊C4—22no—
313C2×C4⋊C4—24no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—22no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—22no—
318C2×C4⋊C4—24no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—24no—
322C2×C4⋊C4—24no—
323C2×C4⋊C4—21no—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—22no—
326C2×C4⋊C4—21no—
327C2×C4⋊C4—84no—
328C2×C4⋊C4—84no—
329C2×C4⋊C4C2×D411yescharacteristic
330C2×C4⋊C4C2×D411yescharacteristic
331C2×C4⋊C4C2×D411yescharacteristic
332C2×C4⋊C4C2412yes—
333C2×C4⋊C4C2414yes—
334C2×C4⋊C4C2411yescharacteristic
335C2×C4⋊C4C2412yes—
336C2×C4⋊C4C2411yescharacteristic
337C2×C4⋊C4C2411yescharacteristic
338C2×C22⋊C4—22no—
339C2×C22⋊C4—24no—
340C2×C22⋊C4—22no—
341C2×C22⋊C4—22no—
342C2×C22⋊C4—22no—
343C2×C22⋊C4—22no—
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—22no—
347C2×C22⋊C4—22no—
348C2×C22⋊C4C2×D412yes—
349C2×C22⋊C4C2×D412yes—
350C2×C22⋊C4C2×D412yes—
351C2×C42—21no—
352C2×C42—22no—
353C2×C42—21no—
354C2×C42—22no—
355C2×C42—22no—
356C2×C42—44no—
357C2×C42C2×D411yescharacteristic
358C2×C42C2×D411yescharacteristic
359C2×C42C2×D411yescharacteristic
360C2×C42C2×D412yes—
361C2×C42C2×D412yes—
362C2×C42C2×D411yescharacteristic
363C2×C42C2411yescharacteristic
364C2×C42C2412yes—
365C2×C42C2411yescharacteristic
366C2×C42C2411yescharacteristic
367#107—24no—
368#107—28no—
369C4⋊C8—24no—
370C4⋊C8—44no—
371C4⋊C8—42no—
372#103—24no—
373#103—28no—
374#103—24no—
375#103—42no—
376#103—44no—
377#103—42no—
378#102—28no—
379#102—28no—
380#102—28no—
381C22⋊C8—28no—
382C4×C8—24no—
383#95—21no—
384#95—22no—
385#95—22no—
386#95—21no—
387#95—21no—
388#95—21no—
389#95—21no—
390#95—22no—
391#95—21no—
392#95—22no—
393#95—22no—
394#95—22no—
395#95—21no—
396#95—24no—
397#95—24no—
398#95C2×D412yes—
399#95C2×D412yes—
400#95C2×D412yes—
401C24—22no—
402C24C22×D412yes—
403#41—416no—
404#41—88no—
405C2×Q8—21no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8—22no—
411C2×Q8—22no—
412C2×Q8—22no—
413C2×Q8—22no—
414C2×Q8—44no—
415C2×Q8—82no—
416C2×Q8—82no—
417C2×Q8C22×D412yes—
418C2×D4—28no—
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—24no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—24no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—81no—
430C2×D4—81no—
431C2×D4—82no—
432C2×D4—82no—
433C2×D4—82no—
434C2×D4—82no—
435C2×D4#12114yes—
436C2×D4C22×D414yes—
437C22×C4—24no—
438C22×C4—24no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—24no—
442C22×C4—24no—
443C22×C4—24no—
444C22×C4—28no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—21no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—24no—
451C22×C4—42no—
452C22×C4—42no—
453C22×C4—42no—
454C22×C4—48no—
455C22×C4—44no—
456C22×C4—44no—
457C22×C4—82no—
458C22×C4—82no—
459C22×C4—82no—
460C22×C4—82no—
461C22×C4—82no—
462C22×C4—82no—
463C22×C4—84no—
464C22×C4—84no—
465C22×C4C22×D412yes—
466C22×C4C22×D411yescharacteristic
467C22×C4C22×D412yes—
468C22×C4C22×D411yescharacteristic
469C22×C4C22×D414yes—
470C22×C4C22×D411yescharacteristic
471C22×C4C22×D412yes—
472C22×C4C22×D411yescharacteristic
473C22×C4C22×D411yescharacteristic
474C22×C4C22×D411yescharacteristic
475C22×C4C2×#4114yes—
476C22×C4C2511yescharacteristic
477Q16—48no—
478SD16—88no—
479M16—88no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—24no—
483C2×C8—22no—
484C2×C8—22no—
485C2×C8—24no—
486C2×C8—22no—
487C2×C8—22no—
488C2×C8—24no—
489C4⋊C4—24no—
490C4⋊C4—22no—
491C4⋊C4—21no—
492C4⋊C4—22no—
493C4⋊C4—24no—
494C4⋊C4—22no—
495C4⋊C4—24no—
496C4⋊C4—22no—
497C4⋊C4—24no—
498C4⋊C4—22no—
499C4⋊C4—42no—
500C4⋊C4—44no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—48no—
504C4⋊C4—44no—
505C4⋊C4—44no—
506C4⋊C4—48no—
507C4⋊C4—42no—
508C4⋊C4—42no—
509C4⋊C4—44no—
510C4⋊C4—42no—
511C4⋊C4—81no—
512C4⋊C4—84no—
513C4⋊C4—84no—
514C4⋊C4—81no—
515C4⋊C4—82no—
516C4⋊C4—82no—
517C4⋊C4#12114yes—
518C4⋊C4#13714yes—
519C4⋊C4C22×D412yes—
520C22⋊C4—28no—
521C22⋊C4—24no—
522C22⋊C4—24no—
523C22⋊C4—48no—
524C22⋊C4—44no—
525C22⋊C4—44no—
526C22⋊C4—44no—
527C22⋊C4—82no—
528C22⋊C4—82no—
529C22⋊C4—82no—
530C22⋊C4—82no—
531C22⋊C4—82no—
532C22⋊C4#12118yes—
533C42—24no—
534C42—22no—
535C42—22no—
536C42—24no—
537C42—22no—
538C42—22no—
539C42—42no—
540C42—42no—
541C42—41no—
542C42—44no—
543C42—42no—
544C42—82no—
545C42—82no—
546C42#13714yes—
547C42C22×D412yes—
548C42C2×#4112yes—
549C23—22no—
550C23—22no—
551C23—22no—
552C23—22no—
553C23—22no—
554C23—22no—
555C23—24no—
556C23—42no—
557C23—42no—
558C23—42no—
559C23—42no—
560C23—42no—
561C23—42no—
562C23C2×#12114yes—
563C23C23×D411yescharacteristic
564Q8—24no—
565Q8—42no—
566Q8—42no—
567Q8—42no—
568Q8—42no—
569Q8—42no—
570Q8—164no—
571D4—28no—
572D4—28no—
573D4—44no—
574D4—48no—
575D4—84no—
576D4—88no—
577D4—88no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—21no—
585C2×C4—22no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—28no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—48no—
601C2×C4—48no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—42no—
608C2×C4—44no—
609C2×C4—42no—
610C2×C4—42no—
611C2×C4—44no—
612C2×C4—41no—
613C2×C4—42no—
614C2×C4—41no—
615C2×C4—42no—
616C2×C4—42no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—42no—
620C2×C4—42no—
621C2×C4—41no—
622C2×C4—42no—
623C2×C4—44no—
624C2×C4—88no—
625C2×C4—164no—
626C2×C4—164no—
627C2×C4—168no—
628C2×C4C2×#12112yes—
629C2×C4C2×#12114yes—
630C2×C4C2×#12112yes—
631C2×C4C2×#13712yes—
632C2×C4C2×#13712yes—
633C2×C4C2×#13714yes—
634C2×C4C2×#13712yes—
635C2×C4C2×#13712yes—
636C2×C4#57612yes—
637C2×C4C23×D411yescharacteristic
638C2×C4C22×#4111yescharacteristic
639C2×C4#58512yes—
640C2×C4#58512yes—
641C8—42no—
642C8—44no—
643C8—42no—
644C22—24no—
645C22—24no—
646C22—24no—
647C22—42no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—44no—
652C22—44no—
653C22—44no—
654C22—82no—
655C22—82no—
656C22—82no—
657C22—82no—
658C22C22×#12111yescharacteristic
659C22#348511yescharacteristic
660C22#350011yescharacteristic
661C22C22×#13711yescharacteristic
662C22C22×#13711yescharacteristic
663C22C2×#57611yescharacteristic
664C22#358511yescharacteristic
665C4—22no—
666C4—22no—
667C4—22no—
668C4—24no—
669C4—22no—
670C4—24no—
671C4—42no—
672C4—42no—
673C4—41no—
674C4—42no—
675C4—82no—
676C4—84no—
677C4—81no—
678C4—82no—
679C4—81no—
680C4—82no—
681C4#303914yes—
682C4#348512yes—
683C4#352112yes—
684C2—44no—
685C2—84no—
686C2#3600911yescharacteristic
687C2C2×#303911yescharacteristic
688C2#6042811yescharacteristic
689C2#6171911yescharacteristic
690C2#6172611yescharacteristic
691C2#6253711yescharacteristic
692C2#6259411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#550C22
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #36180C2semidirect
4. #29869C2semidirect
5. #29717C2semidirect
6. #28716C2semidirect
7. #28577C2semidirect
8. #28160C2semidirect
9. #27644C2semidirect
10. #25322C2semidirect
11. #25223C2semidirect
12. #24682C2semidirect
13. #24548C2semidirect
14. #23679C2semidirect
15. #23630C2semidirect
16. #22700C2semidirect
17. #22502C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#33332
#361802
#343302
#298692
#297172
#287162
#285772
#281602
#276442
#253222
#252232
#246822
#245482
#236792
#236302
#227002
#225022